Sign, then Ratify : Negotiating under Threshold Constraints

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Sign, then Ratify : Negotiating under Threshold Constraints

Sylvie Thoron

To cite this version:

Sylvie Thoron. Sign, then Ratify : Negotiating under Threshold Constraints. 2006. �halshs-00410841�

GREQAM

Groupement de Recherche en Economie Quantitative d'Aix-Marseille - UMR-CNRS 6579 Ecole des Hautes Etudes en Sciences Sociales

Universités d'Aix-Marseille II et III

Document de Travail n°2006- 50

Sign, then Ratify:

Negotiating under Threshold Constraints

Sylvie THORON

December 2006

Sign, then Ratify:

Negotiating under Threshold Constraints ^{1 2}

Sylvie Thoron ^b December 2006

Abstract :

The procedure for implementing any international treaty necessarily involves two steps. The negotiation phase which culminates in the signature of the treaty is followed by a ratification phase. This last phase is governed by a rule which determines how far the ratification process has to advance before the treaty can come into effect. The purpose of this paper is to analyse, using a game theoretical approach, the possible consequences of this minimum participation rule for the ratification phase and for the negotiation phase. I consider the case of International Environmental Agreements in which, during the negotiation phase, the different parties have to decide on the level of a global target and on how to share the efforts necessary to reach it.

I use a cooperative approach to define what is called the threshold value (T-value). For a given coalition of parties, the T-value gives the expected outcome of the negotiation over sharing a global target, when the parties take into account the minimum participation rule.

Given this T-value, I use a non-cooperative approach to determine which coalition will sign the agreement and what will be its global target. The minimum participation constraint has in fact no impact on the ratification phase because it is always better to refuse to sign rather than to sign and then refuse to ratify. However, I show that the minimum participation constraint can modify the outcome of the negotiation phase. Indeed, it plays a role in a mechanism which can be used by a coalition to signal its leadership commitment. I analyse the conditions under which, at the equilibrium, the leading coalition can provoke an expansion of the signing coalition.

Keywords: International Environmental Agreements. Negotiation. Minimum participation rule. Shapley value. Ratification. Leadership commitment.

JEL Classification C71, C72, D74, K33, N5

b GREQAM, Université de Toulon, France, thoron@univmed.fr

1

Comments and suggestions on a previous version of this paper by Eric Maskin and Hervé Moulin are gratefully acknowledged. I also benefited a lot from discussions with Alan Kirman. Finally, I would like to thanks Jana von Stein for her helpful comments.

2

While this paper has been written, I was a visitor at the Institute for Advanced Study in Princeton and at the

Department of Economics at Princeton University. I want to thank the Department for its hospitality and the

When Martina Navratilova announced her retirement from tennis a journalist asked her:

« - When you retire, will you still be involved in tennis? »

« - I will not just be involved, I am committed to tennis. »

« - What is the di¤erence? »

« - Think of ham and eggs.

The chicken is involved but the pig is committed. »

1 Introduction

Contrary to popular belief, international treaties are not negotiated, then signed and then enter into force. The situation is, in fact, more complicated, partic- ularly in the case of multilateral agreements. In accordance with the 1969 Vienna Convention on treaty law, the procedure for implementing any interna- tional treaty must involve two steps. The …rst or negotiation phase culminates in the signature of the treaty by the di¤erent parties. In the particular case of the Kyoto protocol, for example, the parties had to agree on the amount by which total greenhouse gas emissions should be reduced and to decide on how to split the e¤orts necessary to achieve this amount. The second or rati…cation phase requires the rati…cation by the appropriate national bodies who thus give the o¢cial consent of each country to the agreement. When the agreement is multilateral, there may be special requirements specifying what stage in the rat- i…cation process has to be reached before the agreement enters into force. The rati…cation phase is then governed by a minimum participation constraint which plays the role of a threshold. Depending on the treaty in question, this threshold is de…ned in very di¤erent ways: as a set of countries, a number of countries or a percentage of the targeted polluting substances (cf. Barret (2003) pages 165- 195). Sometimes, the rule is a combination of the three. In the particular case of the Kyoto protocol, two conditions had to be met before the agreement came into e¤ect. The …rst rule was that at least 55 countries which originally signed had also to have rati…ed. The second rule required that the countries which had rati…ed must have accounted for at least 55% of the pollution produced in the reference year (1990).

The rule governing the implementation of an international agreement is therefore nothing other than the imposition of a threshold which must be met before the treaty comes into e¤ect. The question I address in this paper con- cerns the consequences of this minimum participation constraint or threshold for the rati…cation phase and for the negotiation phase.

Such thresholds are often invoked in the literature on games of contribu-

tion to a public good, for example. It is argued there that they are incentive

compatible (see for example the mechanisms proposed by Bagnoli and Lipman

(1989), Palfrey and Rosenthal (1984), or the experimental evidence presented in

a survey by Ledyard (1995)). What the negotiators are seeking when they de-

cide on the rule which will govern the rati…cation phase is to induce the highest

number of countries to participate in the agreement. In a book which describes the negotiations leading to the Montreal Protocol on protection of the ozone Layer, Ambassador Richard Benedick (1998) explains that the U.S. started by proposing a very high threshold:

"There was concern that the United States could, in a situation analogous to its unilateral 1978 action, …nd itself bound to the obligations of an "inter- national" protocol while its ma jor competitors were not. As a legacy of the domestic debate, some U.S. agencies insisted on pushing for a proportion of consumption of 90 percent or higher as the trigger for entry into force and other actions."

But as a the result of the negotiation:

"An inevitable, and reasonable, compromise was struck in Montreal, provid- ing that entry into force would require rati…cation by at least 11 parties, together constituting at least two-thirds of estimated global consumption of controlled substances as of 1986 (article 16)[...] Most observers believed that this would provide a su¢cient critical mass to increase the pressure on any potential large holdouts to join the treaty."

Indeed, in the political science literature, di¤erent arguments can be found, to sustain the idea that the decision to ratify is, in part, driven by the actions of other states: if other governments are ratifying en masse, governments have an incentive to ratify as well (von Stein (2006)). An argument developed by Finnemore and Sikkink (1998) is that at a certain "tipping point" in a norm’s evolution, a "norm cascade" takes place, and once rati…cation becomes the norm, states ratify in large numbers because of pressure from other states and non-state actors. However, these di¤erent aguments do not take into account the simple fact that the countries which have to ratify have already signed the agreement.

To say that they have to take a decision again about their participation when they have to ratify amounts to saying that there is a strategic exploitation of this two step procedure. Some countries may have an incentive to sign, in order to encourage the others to do the same, whilst knowing that they themselves will not ratify in order to free ride. Will they always do this? In the model developed in this paper it will be the case that it is better not to sign rather than to sign and not ratify.

However, when Benedick speaks about a pressure on any potential large

holdouts to join the treaty, this can be understood in a di¤erent way. This does

not necessarily refer to the rati…cation process but can concern the paticipation

in the …rst negotiation phase and signature. I will focus on the problem of

participation, analyzing the consequences of the minimum participation rule

during the negotiation. I will claim that, in the case of multilateral agreements,

the di¤erent parties take the rule of the rati…cation phase into account while

they are negotiating their contributions. The negotiators clearly understand

that they can use the minimum participation constraint strategically. With regard to the 90 percent proposal of the United States at Montreal, Benedick (1998) says:

"Many observers feared that such a requirement could hold the treaty hostage to Japan or the Soviet Union, which might then weaken the protocol by extract- ing other concessions as a price for adherence."

Barret’s (2003) interpretation of this observation is that these countries could exploit a pivotal position to increase their weight in the negotiation. He says (page 319):

"So why was the two-thirds hurdle preferable to the 90 percent threshold?

There are, I think, two reasons. The …rst is that the 90 percent threshold would have given bargaining power to the USSR and Japan. If the two-thirds hurdle were satis…ed and the treaty entered into force, then it probably would have been in the interests of the USSR and Japan to join [...] However, if the 90 percent threshold were not satis…ed, entry by the USSR and Japan would decide whether the treaty entered into force [...] Hence, [these countries] might have used the higher threshold to obtain concessions."

Therefore, he considers that the bargaining power of the di¤erent parties to the negotiation is measured by their voting power, if we interpret the rati…cation as a vote. However, if this were the case, the same argument should be applied to the big CFC-consumers. The European Union and the United States had the greatest bargaining power, in both respects. Did these two parties try to weaken the treaty nevertheless? In fact the opposite was true, the U.S. and the EU were considered as the leaders in the negotiation and their high …nal contributions to the treaty do not support this argument.

My claim is that the di¤erence between the U.S. and the E.U. on one hand and the USSR and Japan on the other hand was not that the latter had more bargaining power than the former but that the U.S. and the E.U. were more committed. The way in which the threshold is de…ned can be interpreted as a signal about the degree to which the di¤erent countries are committed. Of course, this depends on the kind of rule considered. When the minimum partic- ipation rule is de…ned as a number of countries there is no way, for one country or the other, to play a speci…c role and the threshold cannot be used as a signal about commitment. At the other extreme, when the rule is a list of countries, it is clear that these countries are committed to the agreement. In the case of the rule used in the Montreal protocol or the Kyoto protocol there is a signal but one which is noisy.

Another signal about a country’s commitment is its contribution to the

agreement. Heike Schröder was an o¢cial note maker during the negotiations

of the Kyoto protocol. Commenting on the result of the negotiation about the reduction commitments she reports that:

"With 8 per cent of 1990 levels, the EU retains its leadership commitment, albeit on a very narrow margin. The US consented to a 7 per cent reduction target, which is a substantial commitment given its increase in emissions since 1990 of about 23 per cent." (Schröder (2001) p. 79 and 80)

But what is the link between these two sorts of signals? In this paper I give a game theoretical explanation of the e¤ect the existence of a given threshold.

I show that it can modify the result of a negotiation in terms of the di¤erences between the countries’ contributions and also in terms of the level of partici- pation. The intuition is the following. Once the countries which have rati…ed satisfy the minimum participation constraint, the agreement is implemented by those countries. At this point, the countries which will ratify later will bene…t from this implementation by the others. Indeed, the cost of implementing the agreement is much higher for the initiating countries ¹ . If it is likely that one country will be among the …rst to ratify, it is also likely that this country will bear a higher cost in implementing the agreement. In other words, the leader- ship commitment mentioned by Schröder can be understood in a literal sense.

As a consequence, such a country will enter into the negotiation with a generous attitude.

If the interpretation I have given makes sense, we should …nd a relationship between a country’s position with respect to the threshold rule (is it necessary or not for example) and its involvement in the negotiation or its willingness to con- tribute to the agreement. Unfortunately, the latter is very di¢cult to measure.

However, the Kyoto protocol constitutes a case which is interesting and unique from this point of view. First, its minimum participation constraint is a rather sophisticated double rule. The …rst rule which speci…es a number of countries is the most common rule applied in environmental treaties. The second rule which speci…es a volume of the polluting substances targeted is much rarer, and it is even more unusual to have a combination of this rule with another rule.

As far as I know, the Montreal protocol is the only other environmental agree- ment with a double rule of this sort (see Barret’s list (2001) pages 165-195 and the site of the Environmental Treaties and Resource Indicators). Furthermore, the Kyoto protocol had another characteristic, which is that the countries have decided to …x not only a global target but also individual targets, called the Assigned Amounts. These assigned amounts and the way they di¤er from the global target can be considered as a proxy for the di¤erent countries’ involve- ment in the negotiation. The 5.5% decrease of global greenhouse gas emissions is shared in a very unequal way (see Table 1 in Section 2). For example, in terms of its own emissions, the contribution of the EU (responsible for 24%

of total CO 2 emissions) is -8%, the contribution of the US (36% of total CO 2

See for example the paper by G. Heal (1993) in which he gives a cost side analysis.

emissions) is -7% while the contribution of Russia (17.5% of total CO 2 emis- sions) is 0% and Australia, which is responsible for just a little more than 2%

of emissions can increase its pollution by 8 percent. Di¤erent justi…cations, historical, technological, and so forth, for these inequalities can be found in the environmental literature. However it is worth noting that the biggest polluters which are necessary to reach the threshold, are the biggest contributors.

How can we analyze this ex-ante e¤ect of a threshold on the result of a negotiation? Here, I adopt a cooperative approach. Cooperative game theory proposes di¤erent tools, solution concepts, to share a worth or a cost based on di¤erent principles. For example, the Shapley value is based on the principle that each player is remunerated according to its incremental worth, that is, to its contribution to the contribution of a given coalition. Then, for a given player, her Shapley value is her expected incremental contribution, when it is assumed that the di¤erent orders in which the partners join the coalition are equally probable. The Shapley value has been proposed "to evaluate the players’

prospects"- Hart and Kurz (1983) p1047. Let me also quote Shapley (1953 p.

307):

"At the foundation of the theory of games is the assumption that the players of a game can evaluate, in their utility scales, every "prospect" that might arise as a result of a play. In attempting to apply the theory to any …eld, one would normally expect to be permitted to include, in the class of "prospects,"

the prospect of having to play a game. The possibility of evaluating games is therefore of critical importance."

The idea that a value can be interpreted as the expected outcome of a nego- tiation has been explicitly developed by Hart and Kurz. Since then, there have been di¤erent attempts in the non cooperative literature to prove what was orig- inally just an interpretation: that the Shapley value can be considered as the ex- pected outcome of a negotiation (see for example Perez-Castrillo and Wettstein (2001) or Maskin (2004)). Indeed empirical studies had already corroborated this interpretation (see for example Littlechild and Thompson (1979)). In this paper, I will not try to propose a non-cooperative game to describe the negoti- ation. I will adopt the cooperative approach keeping in mind the interpretation that the value corresponds to an evaluation of players’ prospects.

Another important aspect of the interpretation of the Shapley value concerns the weights of the di¤erent players in the negotiation. The value is de…ned for a given cooperative game (N; v) and v (N ) is then the amount the players have to share. In our speci…c framework v (N ) can be for example the cost of pol- lution control and, in general, is the total cost of implementing the agreement.

The coalitional function v represents the contribution to this cost of every coali-

tion. However, it cannot re‡ect all the elements which could play a role in the

negotiation. These elements are incorporated in the weights that the di¤erent

players have in the negotiation. Kalai and Samet (1988) say that: "The weights

should be determined by considering such factors as bargaining ability, patience rates, or past experiences". In the Shapley value, everything is considered to be incorporated in the coalitional function and these weights are therefore symmet- rical. By contrast, a weighted Shapley value is de…ned for a vector of exogenous weights. Kalai and Samet (1985) have proposed an axiomatization of the family of weighted Shapley values. Hart and Kurz (1983) have proposed a coalition structure value which represents the Shapley value players can expect when they form coalitions. Intuitively, this is what the players can expect when they form coalitions to increase their weight in the negotiation: "Our view is that the reason coalitions form is not in order to get their worth, but to be in a better position when bargaining with the others on how to divide the maximal amount available." (p. 1052). In this paper, I will consider that the di¤erent countries have di¤erent weights because some of them want to be leaders in the negotiation. Whilst, in the coalition structure value, players form coalitions to increase their weights in the negotiation, here countries form a leading coalition to decrease their share of the gain or to increase their contribution to the cost.

Why should they behave in this way? Because, by doing so, they can increase the level of participation, convincing additional partners to sign a more favor- able agreement. This aspect cannot be understood in a cooperative framework.

In the second part of this paper, I use a non-cooperative approach to describe the agreement formation.

In Section 2, I propose a simple environmental cooperative game with three countries, to illustrate how the minimum participation constraint can mo dify the expected outcome of the negotiation. The threshold value ( T -value), which is de…ned for any cooperative game and for any given threshold is then formally presented in Section 3. The T-value is then used to represent the expected out- come of an environmental agreement negotiation when a threshold govern the rati…cation phase. Section 4 presents three di¤erent non-cooperative models of agreement formation. In the simplest game the countries only have to decide whether to participate in the agreement. A rati…cation phase is introduced in the second game. It is shown that, under mild assumptions, the rati…cation phase is always completed. In the third game a coalition can use the minimum participation rule to signal its leadership commitment. Then, the T-value rep- resent the situation in which the leading coalition is willing to contribute more in order to provoke an expansion of the agreement. Section 5 concludes.

2 How should the e¤orts to reach a global target be shared?

2.1 Who gets the best deal?

In economic theory in general and in cooperative game theory in particular, we

are used to analyse sharing rules. In real environmental treaties the problem is

more complex since the outcome of the negotiation is generaly a global target,

a time table and various rules which are common to every party. However the Kyoto protocol is particular from this point of view since it gives country-speci…c targets. Therefore, it is possible to measure the di¤erent parties’ involvement in the protocol comparing the di¤erent speci…c targets. In Table 1 it can be seen that the biggest parties to the Kyoto protocol were assigned rather stringent targets, whereas smaller countries got more lenient targets. The size of the di¤erent parties is here measured by their volume of green house gas emmissions.

country gg emissions percent total target

Australia 448,71 0,026293337 1,08

Canada 663,39 0,038873074 0,94

Iceland 3,17 0,000185754 1,1

Japan 1351,83 0,079214018 0,94

New Zealand 65,66 0,00384752 1

Norway 52,78 0,003092782 1,01

Russian Federation 1951,81 0,114371417 1

Switzerland 51,5 0,003017777 0,92

Ukraine 577,18 0,033821373 1

United States 6614,85 0,387614456 0,93

EU25 5261,47 0,308309611 0,923

Figure 1 plots Country-speci…c targets in the Kyoto protocol against the percentage of the parties’ total green house emissions. The relationship appears to be decreasing.

This does not correspond to what the theory usually predicts nor to the normal intuition in this kind of situation. The usual idea is that biggest players get the best “deal” because their participation is essential if treaty is to enter into force. Why does this interpretation of bargaining power not correspond to what we observe in the case of the Kyoto protocol? In this Section, I will use a cooperative approach to show how the minimum participation constraint can modify the sharing rule. My claim is that, the de…nition of the threshold necessary for a treaty to enter into force can be interpreted as a signal about how committed di¤erent countries are to the treaty. In the case of the Kyoto protocol, does the fact that the threshold was de…ned as a volume of pollution produced mean that the biggest actors were more committed? We know that the European Union played a key role (see for example the book by Gupta and Grubb (2000) Climate Change and European Leadership). The case of the USA is more controversial. The fact that they refused to ratify seems to show a posteriori that they were not committed. However, as Schröder (2001) explains in her report on the negotiations, the USA were rather active during the discussion about the assigned amounts. She says (page 79):

"On 8 December, US Vice President Al Gore made a 12-hour trip to the

Kyoto Confrence where he instructed his delegation "to show increased negoti-

ating ‡exibility if a comprehensive plan can be put into place". This message

may have swayed US negotiators’ willingness to accept higher targets".

Figure 1:

2.2 The Shapley value of the environmental game

To illustrate the ex-ante e¤ect of the minimum participation rule on the result of the negotiation, I propose to consider an environmental agreement as a cost sharing game. Let the set of players be a set of three countries N = f 1; 2;3 g ; which want to decrease their total pollution: T P = 610 units by 10 percent.

This is what will be called the global target. Assume that the marginal cost of decreasing pollution is decreasing, the cost of reducing pollution by R units is C (R) = p

R. Therefore, the three countries have to share a cost T C = 7:81.

At the outset, each country i generates a percentage of the total pollution T P.

Let us assume that each county’s pollution is, respectively, P 1 = 360 = 59%T P, P 2 = 160 = 26:23%T P; P 3 = 90 = 14:75%T P: Given the 10 percent target, the characteristic function of the sharing cost game is given as follows:

8 S ½ N ; c (S ) = s

10% X

i2S

P i

Therefore, c(S) represents the cost borne by coalition S when it decreases its pollution by 10% on its own. Obviously, this cost sharing game is sub-additive.

For each country, the cost of reducing its pollution by 10 percent is smaller if the

other countries do the same thing. This simple characteristic function captures

the idea that cooperation of countries increases their e¢ciency in decreasing

pollution. The Shapley value for this game is, for each country, its expected

incremental cost (expected IC), given that the orders in which the countries join the coalition are equally probable. An order r is de…ned on the set of countries N. The set of all possible orders is denoted by R(N): For a given order r 2 R (N), each country i’s ranking is denoted by r i . The Shapley value can then be calculated with the following Table:

Orders

…rst 1 1 2 2 3 3

second 2 3 1 3 1 2

third 3 2 3 1 2 1

Sum Shap. val.

Country 1’s IC 6 6 3.21 2.81 3.71 2.81 24.54 ^24:54 ₆

Country 2’s IC 1.21 1.1 4 4 1.1 2 13.41 ^13:41 ₆

Country 3’s IC 0.6 0.71 0.6 1 3 3 8.91 ^8:91 ₆

' ₁ (v) = 4:03 = 51:6% T C;

' 2 (v) = 2:23 = 28:55% T C ; ' ₃ (v) = 1:48 = 18:95% T C

Note that the contribution increases with the country’s volume of pollution but less than proportionally. Now, suppose that to each country is allocated a reduction in pollution in proportion to its contribution to the cost. This gives:

Country 1’s pollution reduction: ^4:03 _7:8 360 = 31:93 = 8:86%:

Country 2’s pollution reduction: ^2:23 _7:8 160 = 17:46 = 10:9%:

Country 3’s pollution reduction: ^1:48 _7:8 90 = 11:6 = 12:86%:

Here again, note that the biggest country reduces its pollution by the smallest percentage. This is a direct consequence of the cost fonction concavity. The small country bene…ts much more from the cooperation of the big country than the opposite. As a consequence, the big country has a higher bargaining power and gets the better deal in the partnership.

2.3 Rati…cation Process and Thresholds

Now, consider that the game is modi…ed by the introduction of a rati…cation process. Indeed, the decisions taken by the di¤erent countries’ delegates during the international negotiation have to be approved by the national institutions.

Given this new process, di¤erent decisional mechanisms could be considered.

First, consider that the three countries sign an agreement in which it is

speci…ed that the signatories are committed to reach a target of a 10% decrease

of their total pollution emissions and which gives the rule which will govern the

rati…cation process. The rule says that the agreement will be considered to be

rati…ed if and only if the countries which have already rati…ed meet a minimum participation constraint. Furthermore, as soon as the agreement is rati…ed it will be implemented by those who have rati…ed it. What does this mean? Once the threshold is reached, the countries which have rati…ed and form a coalition S negotiate the sharing of the e¤orts needed to reach their own common target.

I will call this an ex-post negotiation. The expected outcome of this ex-post negotiation can be considered to be the Shapley value ' (c S ) ; but it depends on the threshold level speci…ed in the rati…cation rule and on the order in which the three countries ratify.

An extreme case is the case in which the agreement can only be implemented if all the countries have rati…ed. Then, at the end of the rati…cation phase, the three countries are exactly in the same situation as before (cf. sub-section 2.1) and the outcome is the Shapley value ' (c), whatever the order in which the countries ratify.

Now, consider another extreme case in which the only condition is that Country 1, the biggest country, has rati…ed. Then, if it is the …rst country to ratify, it will have to bear the biggest cost of v(1) = 6. If it is the second country to ratify, after country 2 it will bear a cost ' ₁ (c 12 ) = 4:6, after country 3, ' ₁ (c 13 ) = 4:85: If it is the third country to ratify, its cost share is, as in the previous case, its Shapley value.

When the agreement can be implemented as soon as the two bigest polluters have rati…ed, two scenarii can occur. First scenario: countries 1 and 2 are the …rst countries to ratify and the threshold is reached without country 3.

Second scenario: country 3 is the …rst or the second country to ratify and the threshold is only reached when all the countries have rati…ed. We will denote by T (¿ ) = f N ;12 g the set of coalitions of countries which can reach the threshold.

Thus, to each random order of rati…cation corresponds one or the other of the two coalitions. In the …rst scenario countries 1 and 2 have to implement the agreement without country 3. In order to do so, they decide to share the common cost of decreasing their pollution by 10%: c(12) = p

10%(P 1 + P 2 ) = 7:2 using the Shapley value for their two person game. Their contributions are then:

T = fN; 12g ; …rst scenario ' ₁ (v 12 ) = 6 + 3:2

2 = 4:6 ' ₂ (v 12 ) = 4 + 1:2

2 = 2:6

When it is country 3’s turn to ratify, then, its incremental cost is just c(123) ¡ c(12) = 7:81 ¡ 7:2 = 0:61. However note that, in the second scenario, in spite of the threshold, the sharing rule will be ' (c) :

Now, consider the case in which the minimum participation constraint is not

de…ned as a coalition of designated countries but a percentage of pollution. For

example, if this threshold is ¿ ; 60% < ¿ · 65% both the coalition of countries

1 and 2 and the coalition of countries 1 and 3 can reach the threshold and

T (¿ ) = f N ;12; 13 g .

Then, there are three di¤erent scenari. In the orders r = (r 1 = 1 ^{r st} ; r 2 = 2 ^nd ; r 3 = 3 ^rd ) or r ⁰ = (r ⁰ ₁ = 2 ^nd ; r ⁰ ₂ = 1 ^rst ; r ⁰ ₃ = 3 ^rd ) countries 1 and 2 ratify

…rst and implement the agreement. Their contributions are:

' 1 (c 12 ) = 4:6 and ' 2 (c 12 ) = 2:6:

In the orders e r = (r 1 = 1; r 2 = 3; r 3 = 2) and e r ⁰ = (r 1 = 3; r 2 = 1; r 3 = 2), countries 1 and 3 ratify …rst and implement the agreement. Their target is C (13) = 6:7 and their contributions:

' ₁ (c 13 ) = 4:85 and ' ₃ (c 13 ) = 1:85:

Lastly, in the orders b r = ( r 1 = 3; r 2 = 2; r 3 = 1); b r ⁰ = (r 1 = 2; r 2 = 3; r 3 = 1) the agreement is only implemented when the three countries have rati…ed and the contributions correspond to the Shapley value.

However, the sequentiality of decisions described here is not consistent with what happens in the case of a real environmental agreement.

2.4 Putting the Cart before the Horse

In the case of a real environmental agreement, the sharing of the e¤orts necessary to reach the global target is decided prior to the rati…cation process, before the countries know which of them will be active …rst. Therefore, let us modify the previous game as follows. The objective of the international negotiation is to decide how to share the e¤orts needed to reach the global target, given that it will be followed by a rati…cation phase. I call this an ex-ante negotiation. I make two assumptions about the rati…cation process and the countries’ conjectures about how it will go.

First, I assume that rati…cation always occurs. In the long term, all the countries will have rati…ed. However, in our framework there is no explicit representation of timing. The only aspect which matters is the order in which the di¤erent countries ratify. This matters because a country is not in the same position when it has to start the implementation of the agreement or when it can join a group of countries which have already started. Of course this is only true if there is a delay between the time the agreement enters into force and the time at which the additional country rati…es. Therefore, the assumption is that this delay is enough for the …rst countries to bear a higher cost of implementing the agreement and for the additional country to bene…t from the decrease in marginal cost.

The second assumption is with regard to the conjectures of the countries’

delegates who participate in the negotiation. I assume that the participants in

the international negotiation have no idea about the order in which the di¤er-

ent countries will ratify. This is not a strong assumption since the delegates

who participate in the international negotiation are not the same individuals as

those who participate in the national process of rati…cation. The delay between

the two phases is another justi…cation for this assumption. Therefore, all the

countries will ratify but the delegates do not know in which order and they

attribute the same probability to each one of these orders. We can …nd di¤erent arguments along these lines in the literature on political science.

However the negotiating parties know the rule which will govern the rat- i…cation process. First note that, without thresholds the situation is similar to that we described in subsection 2.1, where we presented the game without a rati…cation process. The sharing rule decided during the international nego- tiation is just postponed until the end of the rati…cation process. The same thing is true if the threshold is so high that it can only be reached when the three countries have rati…ed. Here, this means that the threshold ¿ is such that ¿ > 75%T P. Therefore, in this situation the expected outcome of the negotiation is the Shapley value ' (c).

The question is now to know what happens when a binding threshold is introduced. Then, as we saw in 2.2, the outcome of an ex-post negotiation depends on the order of rati…cation. Now, let us consider the expectation of the outcome of the ex-post negotiation, given that the di¤erent possible orders of countries ratifying are equally probable. My claim is that this is the outcome of the ex-ante negotiation and this de…ne a value for the cooperative game (N; c) and for the given threshold. I will denote by Á (c; T ) what I will call the threshold value. This value can be interpreted as follows: It is the delegates’

"prospects" in the ex-ante negotiation when the outcomes of what would be ex-post negotiations are common knowledge. It does not mean that these ex- post negotiations will occur. It means that the countries use what would be the outcome of an ex-post negotiation as an argument in the ex-ante negotiation.

I will de…ne the T-value precisely in the following sections but let us see what happens in our example.

The …rst case I will consider is the case in which the threshold is high. That is when:

T = f N; 12 g or 65% < ¿ · 75%;

In that case we saw in the previous sub-section that in the …rst scenario, that is in two orders out of six, countries 1 and 2 contribute more than their Shapley value, respectively, ' ₁ (c 12 ) = 4:6 > 4:08 and ' ₂ (c 12 ) = 2:6 > 2:23.

On the other hand, country 3 contributes much less than its Shapley value c (123)¡c(12) = 0:61 < 1:48. In the other scenario the three countries contribute their Shapley value. Therefore, the threshold value can be calculated as shown in the following table:

Orders

…rst 1 1 2 2 3 3

second 2 3 1 3 1 2

third 3 2 3 1 2 1

Sum T-value 1’s S-value ² 4.6 4.03 4.6 4.03 4.03 4.03 24.54 ^24: ₆ ⁵⁴ 2’s S-value 2.61 1.1 2.61 3 1.1 3 13.41 ^13: ₆ ⁴¹ 3’s S-value 0.6 1.85 0.6 2 1.85 2 8.91 ^8: ₆ ⁹¹

2

The Shapley value is calculated, for Country 1 in each given order.

Á 1 (c; ¿) = 4 ¤ 4:08 + 2 ¤ 4:6

6 = 4:25

Á ₂ (c; ¿) = 4 ¤ 2:23 + 2 ¤ 2:6

6 = 2:35

Á ₃ (c; ¿) = 4 ¤ 1:48 + 2 ¤ 0:61

6 = 0:99

Note that, in each given rati…cation order the global target is reached. However, the contribution of country 3 which may be able to ratify after the implemen- tation of the agreement is less than its Shapley value. Country 3 gains at the expense of countries 1 and 2.

The second case is the case in which the threshold is at an intermediate level.

That is when:

60% < ¿ · 65% or T (¿ ) = f N ; 12; 13 g Then, the threshold value is:

Á ₁ (c; ¿ ) = 2 ¤ 4:6 + 2 ¤ 4:85 + 2 ¤ 4:08

6 = 4:51

Á ₂ (c; ¿ ) = 2 ¤ 2:6 + 2 ¤ 2:23 + 2 ¤ 1:11

6 = 1:98

Á ₃ (c; ¿) = 2 ¤ 1:48 + 2 ¤ 0:61 + 2 ¤ 1:85

6 = 1:31

In that case, countries 2 and 3 gain at the expense of country 1. Country 1, which always participates in the …rst implementation of the agreement, is the biggest contributor.

In the last case Country 1’s participation is the only necessary condition for the agreement to enter into force. The T-value is decribed in the following table:

Orders

…rst 1 1 2 2 3 3

second 2 3 1 3 1 2

third 3 2 3 1 2 1

Sum T-value 1’s order value 6 6 4.6 4.03 4.86 4.03 29.52 ^29:52 ₆ 2’s order value 1.15 1.15 2.61 2.23 1.1 2.23 8.24 ^8:24 ₆ 3’s order value 0.65 0.65 0.6 1.48 1.85 1.48 6.71 ^6:71 ₆

Á ₁ (c; 1) = 4:92 = 63% T C;

Á ₂ (c; 1) = 1:37 = 17:58% T C;

Á ₃ (c; 1) = 1:12 = 14:31% T C

3 The T -value

3.1 Notation and de…nitions

In this sub-section, well known and new concepts and results which will be used in the following Section will be presented. Some of them are new, others are standard. Consider a set of players U: A coalition of players is a subset S ½ U:

A coalitional function game (U; v) is de…ned for a given coalitional function v, which associates a worth v (S) to each coalition S ½ U. I will only consider coali- tional function games which are superadditive, that is, for two disjoint coalitions S and T ½ U, v (S) + v (T ) · v (S [ T). The incremental worth of player i to coalition S; i 2 S is v(S) ¡ v(S n i). A null player is a player i whose incremental worth to each coalition is zero: v(S [ i) ¡ v(S) = 0; 8 S ½ U . A sub-set N ½ U is called a carrier of game (U; v) if 8 i 2 U n N ; i is a null player. The property of superadditivity captures an important characteristic of environmental agree- ments. The interpretation of superadditivity is straighforward and this concept is often used in the literature on agreement formation. Indeed, it is a common idea that cooperation, generating synergies or externalities, can be represented by a superadditive game. I will also use further below the following properties of the coalitional function:

De…nition 1 The coalitional function game (U; v) is concave (convex) if, for each player i 2 U,

v(T) ¡ v(Tni) · (¸)v(S) ¡ v(Sni); 8S ½ T ; i 2 S

A coalition game is concave if, for each player, her incremental worth de- creases when additional players are added to the coalition she joins. A value of the coalition game (U; v) is a function which associates with each player i in U a real number. For example, the Shapley value can be interpreted as a mutual acceptable sharing of v(N) when the players agree in the following principles:

1- Each player must be remunerated at the level of her incremental worth.

2- This incremental worth depends on the group she joins.

3- An average incremental worth must be calculated taking into account the order in which the di¤erent players join the group. The di¤erent orders have the same weight (or are equiprobable).

Formally, for every game in coalitional function form (U; v), the Shapley value can be characterized by three very simple axioms (cf Shapley (1953)).

Axiom 1 E¢ciency and null player. If N ½ U is a carrier of game v; then:

X

i 2 N

' i (v) = v(N )

Axiom 2 Additivity. If (U; w) is de…ned such that w (S) = v(S)+u(S); 8 S ½ U then:

' (w) = ' (v) + ' (u)

Axiom 3 Anonymity. If ¼ is a permutation of U and ¼v de…ned such that

¼v (¼S) = v(S); 8 S ½ U then:

' _{¼ i} (¼v) = ' _i (v)

De…nition 2 8 K µ U; 8 ® 2 R ; a unanimity game ®U K can be de…ned as follows:

8 S ½ U; v(S) =

½ ® if K ½ S 0 otherwise

In an axiomatization proposed by Aumann (2005), Anonymity and Null player are replaced by the following Axiom:

Axiom 4 8K µ U; 8® 2 R; ' i (®U K ) =

½ ®

j K j ; if i 2 K;

0 otherwise

If N is a …nite carrier of game (U; v), the Shapley value for player i can be calculated as her expected incremental value over all possible orders de…ned on N , under the assumption that each order appears with the same probability:

8i 2 N ; ' _i (v) =

P

S ½ N;i 2 S (s ¡ 1)! (n ¡ s)! (v (S) ¡ v (S n i)) n!

For each coalition M ½ N, de…ne a coalitional function v M such that:

v M (S) = v (S \ M ) Then:

8 i 2 M ½ N;

' _i (v M ) = P

S ½ M; i 2 S (s ¡ 1)! (m ¡ s)! (v M (S) ¡ v M (Sni)) m!

In this last case, ' _i (N; v M ) is the value obtained by the members of coalition M ½ N when they share v (M). Now, for each given coalition M ½ N; de…ne a game (N ; v ^¤ _M ), called the incremental game:

8S ½ N ; v ^¤ _M (S) = v(S) ¡ v M (S)

Note that, if S ½ M , v ^¤ _M (S) = 0 and since the game is superadditive, v ^¤ (S) ¸ v(S). The Shapley value for each game (N; v _M ^¤ ) can be de…ned as:

8 i 2 N n M ; ' i (v ^¤ M ) =

P

S ½ N;i 2 S (s ¡ 1)! (m ¡ s)! (v _M ^¤ (S) ¡ v _M ^¤ (S n i)) (n ¡ m)!

= P

S ½ M ;i 2 S (s ¡ 1)! (m ¡ s)! (v (S [ N n M ) ¡ v (S n i [ N n M ))

m!

This is the value obtained by players in N n M ; when they share what can be called, by extension, their incremental value v (N) ¡ v (M ) : In other words, this is the value which is calculated for each N n M -member, assuming that a coalition M got its value v(M ) already and taking into account all possible orders of players in N n M when they join the coalition M .

Example 1 An obvious case is that in which M is a singleton fig. Then, ' _i ¡

v _f i g

¢ = v ( f i g ) and ' _i ³ v _N ^¤ _{n i} ´

= v (N ) ¡ v (N n f i g ) is player i’s incremental value:

Example 2 Consider another case in which N = f 1; 2; 3; 4 g and M = 12:

Then,

' ₁ (v 12 ) = v ( f 1 g ) + v (12) ¡ v ( f 2 g ) 2!

and

' ₃ (v ₃₄ ^¤ ) = v (N ) ¡ v (N n 3) + v (123) ¡ v (12) 2!

The following Proposition proved in a companion paper (Thoron 2006) shows that there is a general relationship between the Shapley values de…ned for a game v; for the games v M ; and for the games v _M ^¤ . Consider C m the class of coalitions M ½ N ; of the same size m:

C m = f M ½ N : #M = m g For each class C m , the following relationship can be proved:

Proposition 1 Let N any …nite carrier of game v. For each given class C m ; m · n; the following relationship exists between the Shapley values de…ned for three categories of games: the original game v, the games v M de…ned for each m ¡ size sub-coalition M ½ N, and the associated incremental games v ^¤ _M :

' _i (v) = m!(n ¡ m)!

n!

2 6 4 X

M 2C

i 2 M

' _i (v M ) + X

M2C

i = 2 M

' _i (v ^¤ _M ) 3 7 5

3.2 Heuristic approach

Following the usual heuristic description of the Shapley value, players have to

meet in a bargaining room to share the value of the grand coalition. They

arrive sequentially and the order in which they do so is determined by chance,

with all arrival orders equally probable. Each player, when she enters the room,

demands and is promised the amount which her participation contributes to the

value of the grand coalition.

Here, I modify this description by introducing two rooms: the waiting room and the bargaining room. The participants arrive sequentially and in a random order in the waiting room. But there is nothing to share before the coalition formed by the players in the waiting room has reached the threshold. When the last player necessary to reach this threshold enters in the waiting room, the door is closed from outside and the usual process is followed by the present players.

Players arrive sequentially and in a random order in the bargaining room. Each player from the waiting room, when she enters the bargaining room, demands and is promised the amount which her participation contributes to the value of the coalition. For each coalition S ½ N ; the value ' _i (v S ) represents what player i 2 S can obtain in this negotiation.

When the waiting room is empty again, the door of the waiting room is reopened and the remaining players arrive sequentially and go straight to the bargaining room in order to demand the amount which their adherence con- tributes to the value of the grand coalition. For each coalition S , the value ' _i (v _S ^¤ ) represents what player i 2 N n S can obtain in this negotiation.

3.3 Characterization

Denote by r (N ) an order de…ned on the set of players. For a given order r(N );

each player i is associated with a ranking r i . For a given leading coalition L ½ N and a given order r(N ); consider the coalitions S ½ N ; which satisfy the three following conditions:

(i) L ½ S

(ii) All the players who belong to coalition S arrive successively in r(N ):

given i 2 S; such that r i = M in k 2 S r k and j 2 S such that r j = M ax k 2 S r k ; 8k 2 S, r j ¸ r k ¸ r i . In other words, think about S as a block in the order r(N):

(iii) One of the S-members arrives …rst in the order. In other words r i = M in k 2 S r k = 1.

For a given leading coalition L ½ N , to each order r(N ) corresponds a unique coalition S; which satis…es the three previous conditions and has the smallest number of members. But each coalition S may correspond to several orders. This de…nes an injective but not surjective application from the set of orders to the set of coalitions L = fS : L ½ Sg . Denote by ® S the number of orders associated with coalition S , this is the number of orders in which the S-members arrive in …rst position but among them the last to arrive is a L- member. In other words, in the order r (N) ; the threshold is reached when the last L-member has arrived and all the players who have a smaller ranking constitute coalition S, associated with this order. Note that as a consequence of the injective application, we have: P

S 2 L ® S = n!

Example 3 For each coalition S 2 T (¿ ), ® S = (n ¡ s)!s!: This is the number of orders in which the S-members arrive in the …rst positions.

Example 4 If T (¿ ) = C s ,again ® S = (n ¡ s)!s! but in that case, the coalitions

in T (¿ ) of larger size do not correspond to any order.

Given this de…nition of weights ® S ; S 2 L, we can de…ne the threshold value as follows:

De…nition 3 For any …nite carrier N of game v and a leading coalition L ½ N , the T -value for each player i 2 N is given by:

Á i (v; L) = P

S 2 L ;i 2 S ® S ' _i (v S ) + P

S 2 L;i = 2 S ® S ' _i (v _S ^¤ ) n!

In a companion paper (Thoron (2006)), I proved that the T-value is charac- terized by the following Axioms:

Axiom 5 E¢ciency and null player. If N is a carrier of game v; then P

i 2 N Á _i (v; L) = v(N )

Axiom 6 Additivity. If w is de…ned such that w (S) = v(S) + u(S) then:

Á i (w; L) = Á i (v; L) + Á i (u; L)

Axiom 7 Anonymity. If ¼ is a permutation of U and ¼v de…ned such that

¼v (¼S) = v(S); then,

Á _{¼ i} (¼v; ¼L) = Á _i (v; L)

Axiom 8 Transfer. For any given a unanimity game ®U K . For any given L µ U :

If K µ L; Á _i (®U K ; L) =

½ ®

jKj ; if i 2 K ; 0 otherwise If L µ K;

½ Á _i (®U K ; L) = 0; 8 i = 2 K;

Á _j (®U K ; L) = Á _i (®U K ; L) + Á _j (®U K ; K ) ; if i 2 L and j 2 K n L This last Axiom is equivalent to the following Axiom:

Axiom 9 For any given K µ U; and ® 2 R ; If K µ L; Á _i (®U K ; L) =

½ ®

j K j ; if i 2 K ; 0 otherwise

If L µ K; Á i (®U K ; L) = 8 >

<

> :

® j K j

jLj

j K j ; if i 2 L;

jKj ®

³

1 + _jKj ^{j L j} ´

; if i 2 K nL;

0 otherwise

What is the meaning of these two last axioms? Note that, for a given unanim-

ity game ®U K , Á _i (®U K ; K) = ' _i (®U K ). When the coalition which constitutes

the threshold is L = K , the T-value coincides with the Shapley value. Given

this, the Transfer Axiom shows how a smaller threshold modi…es the Shapley

value. In a unanimity game, all the players are identical and they are all nec-

essary to realize a worth. In that case, when there is no threshold to bias the

sharing, the value is expected to be the equal sharing of v (K) between the K- members (see Axiom 4 of the Shapley value). Then, when a smaller threshold is introduced, it may attribute di¤erent positions to previously symmetric players.

In that case, the player who is above the threshold has a higher value than the player who is below the threshold. Furthermore, the di¤erence between the two values is just the value without threshold, that is the Shapley value. Therefore Á _i (®U K ; L), which is the share received by the members of the threshold coali- tion L is also what they transfer to the non-member players. For example, if 2 ¤ j L j = j K j , the members of the threshold coalition transfer half of their share to the non members.

A natural idea would be that the players who are below the threshold have a kind of veto power since the sharing cannot be done without them. This is misleading. In a unanimity game ®U K the K -members all have this veto power. The worth ® cannot be realized without each of the K -members. In fact, the threshold introduces an additional distinction. Remember that, because the game is superadditive, for any coalition S it is always better to get its incremental worth rather than its worth: v(K) ¡ v(K n S) ¸ v(S). Therefore, even if all players in K participate in the sharing of v(K ), there is a di¤erence among them depending on their position with respect to the threshold. The players who are below the threshold will not have the possibility to share an incremental value v(K ) ¡ v(K n S) instead of a value v(S). One can say that the players who are below the threshold are more committed to the sharing. They will initiate the sharing. The last axiom measure the price they pay for this.

4 Non-cooperative models of agreement forma- tion

4.1 Incentives

The purpose of the previous Section was to explain how a country can signal

its leadership commitment through the minimum participation constraint. The

question remains as to why a country should behave like this. Consider the

situation in which countries have the possibility to participate or not in an envi-

ronmental agreement. Consider that they anticipate that the expected outcome

of the negotiation will be the Shapley value. They also anticipate that the nego-

tiation will be followed by a rati…cation phase. In this framework, a sub-set of

countries may have the right incentives to sign and ratify the agreement. They

simply bene…t more from participating in the agreement than from the external-

ities from which they could bene…t by staying independent. We say in this case

that the coalition of countries is stable in the sense de…ned by D’Aspremont et

al. (1983). This is also equivalent to say that there exists a Nash equilibrium of

a simple coalition formation game in which the set of strategies is just binary

(see Thoron (1998)). As it is well known from the literature on the theory of sta-

ble cartels, the size of this coalition depends, of course, on the payo¤ function,

but it is often rather limited. In other words, the stable agreement is "small".

However, as Carraro and Siniscalco (1993) have shown, an expansion of this coalition can be induced through transfer and commitment. Transfer without commitment is not enough since, if the transfer is e¢cient and provokes the coalition’s expansion, then the countries which belonged to the initial stable coalition may now have an incentive to leave and free ride to bene…t from the larger externalities and to avoid bearing a higher cost. Therefore, Carraro and Siniscalco propose the following mechanism: The countries belonging to the ini- tial stable coalition commit to cooperate. Once this is done, they choose the transfer in order to maximize the number of signatories:

1) transfers are self-…nanced, i.e. the total transfer must be lower than the gain that the committed countries obtain from expanding the coalition.

2) the move to a larger coalition must be Pareto-improving, i.e. all countries must be better o¤ than in the situation preceding the coalition expansion, and better o¤ than in the case of non-cooperation. (cf. Carraro and Siniscalco (1993) p 316).

The purpose of this Section is to give a formal framework for a similar mech- anism. However here, it is assumed that the transfers are not direct monetary transfers but endogenously emerge from the negotiation as a modi…cation of the sharing rule. Indeed, this is the simplest way to organize these transfers. In other words, the leading coalition is more "‡exible" in the negotiation. I assume that the T -value is the new sharing rule when a leading coalition constitutes a threshold. It represents this leading coalition’s willingness to organize a transfer in order to make an expansion possible. Given this new sharing rule, the ques- tion is to know which leading coalition will form and which expansion will be possible.

There are two degrees of commitment: the strongest degree of commitment is to send the signal that the country will be a leading country. The second degree of commitment is, for a country, to say that it will participate but needs to be subsidized. Some countries are willing to be strongly committed if and only if other countries participate. They only sign the agreement and assume the high commitment if the others sign also. The last possibility is just to refuse to participate.

4.2 Coalitional function

First, I will consider a simple and aggregated game called game G which will be used in the subsequent Sections as a benchmark. Consider a set of n countries N = f1; :::; ng ; each one characterized by an amount of pollution y i ; i 2 N;

in a status quo situation. The implementation of an agreement signed by a subset of countries T means that they choose to reduce their pollution by a percentage x 2 [0; 1] which maximizes their joint net bene…t. The net bene…t is the increase of utility produced by the improvement of the environment less the cost of decreasing pollution. Denote the joint net bene…t of coalition T by P T

¡ x (y i ) _{i 2 T} ¢

in which x is a scalar and Y T = (y i ) _{i 2 T} the vector of the di¤erent

T -members’ pollution: Consider:

b

x T = Arg max

x P T

¡ x (y i ) _i2T ¢

; 8T ½ N

It is assumed that x b i = 0;8i 2 N: The following assumption A1 holds:

A1 : b x T · x b S if S ½ T ; j S j > 1

Example 5 The following payo¤ function is often used in the litterature on en- vironmental agreements (cf. for example Barret (1994), Ray and Vohra (2001), Courtois and Haeringer (2005)...). Consider the symmetric case, y i = y = 1u., 8i 2 N. A coalition of s countries sign an agreement to decrease their pollution by x per cent. The individual bene…t of the aggregate abatment level xsy = xs is:

Q ¡

x (y i ) _{i 2 S} ¢

= Q(xsy)

= axsy ¡ 1

2 x ² s ² y ² = asx ¡ 1 2 x ² s ²

The symmetrical cost of decreasing pollution is, for each one of the s countries:

C (xy) = C (x) = c 2 x ²

Where c is a positive parameter smaller than 4, c · 4. The percentage x is chosen to maximize the aggregate payo¤ of the s countries:

P S ¡

x S (y i ) _{i 2 N} ¢

= P S (xsy)

= µ

asx ¡ 1

2 x ² s ² ¡ c 2 x ²

¶ s

The solution is x b S = _s

^as +c . It can be easily checked that, when c · 4, that is when the marginal cost is not "too high", ^@ _@s ^{x b}

· 0 as soon as s ¸ 2 and Assuption A1 is veri…ed.

Note that, A1 is consistent with another more traditional assumption which is that the pollution abatment increases when the coalition expands. This will be the property of superadditivity of the characteristic function V de…ned as follows:

v (S) = b x S

X

i2S

y i

The worth v(S) is the pollution abatment a coalition S would choose if its

members signed a binding agreement. Note that assumption A1 implies that

characteristic function v is concave: the incremental value of each country is

decreasing. This worth could have been de…ned as a partition function, depend-

ing on a coalition structure. However, it is assumed that only one coalition can

be formed. Implicitly, countries in N nS remain independent and, as speci…ed

above, do not decide any abatment. I will come back to this point below. The

characteristic function V is assumed to be superadditive. Therefore, in particu- lar:

A2 : 8 S ½ N; 8 T ½ S; v (S) ¸ v (S n T )

In a given coalition T , members share the joint bene…ts P T using a sharing rule à (v) to decide about each country’s reduction. Therefore, à _T (v) = (à _i (v)) _{i 2 T} is a vector in which à _i (v) is the percentage applied to country i: The sharing rule is assumed to be e¢cient :

A3 : Ã _T (v) Y T = X

i 2 T

à _i (v) y i = b x T

X

i 2 T

y i = v(T )

As a consequence of superadditivity of v, we can write:

v(T ) = b x T

X

i 2 T

y i ¸ b x T n j

X

i 2 T n j

y i = v(T nj)

But, by Assumption A1:

b x _Tnj X

i 2 T n j

y i ¸ b x T

X

i 2 T n j

y i

Therefore:

b x T

X

i2T

y i ¸ x b T n j

X

i2Tnj

y i ¸ b x T

X

i2Tnj

y i

When an additional country joins the coalition, the coalitional function is such that the total reduction of pollution increases but the reduction by the initial countries decreases. In what follows, we will see that, as a consequence, the initial members bene…t from the new membership through two di¤erent ways.

The vector of status quo (y i ) _{i 2 N} is …xed. Then, the di¤erent countries’

payo¤s are denoted by ¼ i ± ; 8 i 2 N. From this situation, if one agreement is signed by a coalition T ; the T -members choose a pollution abatment v(T) and share their e¤orts to reach this target using as a sharing rule the Shapley value:

' (v). Each T-member is then characterized by the bene…t function drawn from the partnership:

8 i 2 T : P i (T ; ' _T (v))

Note that, in general, the maximization problem depends on what the countries which have not signed the agreement do. This could only been represented by a partition function. However here, this is not necessary since there is only one agreement. The characteristic function is equivalent to a partition function in which the countries in NnT are independent. The payo¤ of these countries need to be de…ned nevertheless. The pollution reduction is an externality they can bene…t from but of course they do not bear the same cost since they do not implement the agreement. If country i does not participate in the agreement, its bene…t from the implementation of the agreement by the T-members is denoted by Q i (T ; ' _T (v)) :

8i 2 N nT : Q i (T ; ' _T (v))

Note that, this means that there is no strategic adaptation of pollution reduc- tion by the non signatories (no leakage). The interpretation is the following:

countries which remain independent do not take into account the quality of the environment. This is why they do not decide any reduction. On the other hand, when other countries decrease pollution, they do not increase or decrease pollu- tion as a best response. Simply, they do not participate in the abatment game.

However, this assumption is not determinant in what follows. It just makes the model easier to write and to understand and do not a¤ect the results. Indeed, we could write explicitly the abatment game between the signatories and the non-signatories without changing the results. In fact here, a non-signatorie’ s payo¤ is just a function of the pollution abatment induced by the signatories.

Therefore, it can also be written Q i (S; ' _S (v)) = Q i (' _S (v) Y S ). The following assumptions hold:

The game is essential: 9 S ½ N : P i (S; ' _S (v)) ¸ ¼ i ± ; 8 i 2 S:

A4 Q i is an increasing function of the abatment:

If S ½ T ; 8 i 2 N n T ; Q i (T ; ' _T (v)) ¸ Q i (S; ' _S (v))

A5 For a given agreement T , the non signatories which do not bear the cost of pollution reduction but can bene…t from it, have a higher payo¤ than the signatories:

If i and j are identical but i = 2 T and j 2 T ; then Q i (T ; ' T (v)) ¸ P i (T; ' T (v))

4.3 Stable agreements

Consider the following agreement formation game called game G. Countries in N have to decide to sign or not to sign an agreement to reduce pollution. For each country i 2 N, its strategy is then denoted by: ¾ i 2 f 0; 1 g . Each strategy pro…le ¾ generates a coalition K = f i : ¾ i = 1 g of countries which sign and implement this agreement to reduce pollution. Payo¤s are then determined as in the previous section. A Nash equilibrium of this simple game ¾ ^¤ generates a '-stable agreement signed by the members of coalition K ^¤ = fi : ¾ ^¤ _i = 1g. This implies that coalition K ^¤ satis…es the two well known conditions of internal and external stability proposed by d’Aspremont et al. (1983) and which must be written in this paper’s framework:

Coalition T ^¤ is internally stable if 8 i 2 K ^¤ : P i (K ^¤ ; ' _K

(v)) ¸ Q i

³

K ^¤ n i; ' _K

ni (v) ´ Coalition T ^¤ is externally stable if 8i 2 NnK ^¤ :

Q i (K ^¤ ; ' _K

(v)) ¸ P i (K ^¤ [ i; ' _K

[i (v))

When the di¤erent countries are identical, that is when 8i 2 N; y i = y,

Q and P only depend on the number of countries which sign the agreement.

Figure 2:

Then, the Shapley is the equal sharing and Q i (K ^¤ ; ' _K

(v)) = Q (k ^¤ ) and P i (K ^¤ ; ' _K

(v)) = P (k ^¤ ) (see Figure 2).

Let us denote by N ^Ã the set of coalitions generated by a Nash equilibrium of game G when the sharing rule is Ã. Then, the members of each one of these coalitions will not have any di¢culty to sign the agreement. When the game is symmetric, Thoron (1998) shows that this equilibrium always exists.

Furthermore, using the concept of Coalition Proof Nash Equilibrium (CPNE), it is shown that this equilibrium is unique. The concept of CPNE is a re…nement of Nash equilibria proposed by Bernheim, Peleg and Wilson (1987), which takes into account coalition deviations and sati…es a property of consistency.

Theorem 1 Game G has one and only one coalition-proof stable cartel, which is the greatest stable cartel.

4.4 Stable agreements with rati…cation phase

The previous game G is one shot: the countries decide symultaneously to sign

or not and the agreement is then implemented. Now assume that this one shot

game, which will be called the negotiation phase, is followed by a rati…cation

phase. In other words, the signatories have to con…rm their participation. This

two-step game will be denoted by G ^r . Assume that each country is then repre-

sented by a team of delegates during the …rst negotiation phase and by domestic